- E0 = 1
- E2 = −1
- E4 = 5
- E6 = −61
- E8 = 1,385
- E10 = −50,521
- E12 = 2,702,765
- E14 = −199,360,981
- E16 = 19,391,512,145
- E18 = −2,404,879,675,441
Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, and/or change all signs to positive. This encyclopedia adheres to the convention adopted above.
The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.
Explicit formulas 
Iterated sum 
An explicit formula for Euler numbers is given by:
where i denotes the imaginary unit with i2=−1.
Sum over partitions 
as well as a sum over the odd partitions of 2n − 1,
where in both cases and
As an example,
E2n is also given by the determinant
Asymptotic approximation 
The Euler numbers grow quite rapidly for large indices as they have the following lower bound
See also 
- Ross Tang, "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series"
- Vella, David C. (2008). "Explicit Formulas for Bernoulli and Euler Numbers". Integers 8 (1): A1.
- Malenfant, J.. "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers". arXiv:1103.1585.